CHAPTER 3 Steady Flows in Pressurised Networks 3.1 MAIN CONCEPTS AND DEFINITIONS The basic hydraulic principles applied in water transport and distribution practice emerge from three main assumptions: 1 The system is filled with water under pressure, 2 that water is incompressible, 3 that water has a steady and uniform flow. In addition, it is assumed that the deformation of the system boundaries is negligible, meaning that the water flows through a non-elastic system. 1 Steady flow Flow Q (m 3 /s) through a pipe cross-section of area A (m 2 ) is determined as Q ϭ ␷ ϫ A, where ␷ (m/s) is the mean velocity in the cross-section. This flow is steady if the mean velocity remains constant over a period of time ⌬t. Uniform flow If the mean velocities of two consecutive cross-sections are equal at a particular moment, the flow is uniform. The earlier definitions written in the form of equations for two close moments, t 1 and t 2 , and in the pipe cross-sections 1 and 2 (Figure 3.1) yield: (3.1) for a steady flow, and: (3.2) for a uniform flow. A steady flow in a pipe with a constant diameter is at the same time uniform. Thus: (3.3)v 1 (t 1 ) ϭ v 2 (t 1 ) ϭ v 1 (t 2 ) ϭ v 2 (t 2 ) v 1 (t 1 ) ϭ v 2 (t 1 ) ٙ v 1 (t 2 ) ϭ v 2 (t 2 ) v 1 (t 1 ) ϭ v 1 (t 2 ) ٙ v 2 (t 1 ) ϭ v 2 (t 2 ) 1 The foundations of steady state hydraulics are described in detail in various references of Fluid Mechanics and Engineering Hydraulics. See for instance Streeter and Wylie (1985). © 2006 Taylor & Francis Group, London, UK Transient flow The earlier simplifications help to describe the general hydraulic behaviour of water distribution systems assuming that the time interval between t 1 and t 2 is sufficiently short. Relatively slow changes of bound- ary conditions during regular operation of these systems make ⌬t of a few minutes acceptably short for the assumptions introduced earlier. At the same time, this interval is long enough to simulate changes in pump operation, levels in reservoirs, diurnal demand patterns, etc., without handling unnecessarily large amounts of data. If there is a sudden change in operation, for instance a situation caused by pump failure or valve clo- sure, transitional flow conditions occur in which the assumptions of the steady and uniform flow are no longer valid. To be able to describe these phenomena in a mathematically accurate way, a more complex approach elaborated in the theory of transient flows would be required, which is not discussed in this book. The reference literature on this topic includes Larock et al. (2000). 3.1.1 Conservation laws The conservation laws of mass, energy and momentum are three fundamental laws related to fluid flow. These laws state: 1 The Mass Conservation Law Mass m (kg) can neither be created nor destroyed; any mass that enters a system must either accumulate in that system or leave it. 2 The Energy Conservation Law Energy E (J) can neither be created nor destroyed; it can only be transformed into another form. 56 Introduction to Urban Water Distribution V 1 V 2 2 1 Figure 3.1. Steady and uniform flow. © 2006 Taylor & Francis Group, London, UK 3 The Momentum Conservation Law The sum of external forces acting on a fluid system equals the change of the momentum rate M (N) of that system. The conservation laws are translated into practice through the applica- tion of three equations, respectively: 1 The Continuity Equation. 2 The Energy Equation. 3 The Momentum Equation. Continuity Equation The Continuity Equation is used when balancing the volumes and flows in distribution networks. Assuming that water is an incompressible fluid, i.e. with a mass density ␳ ϭ m/V ϭ const, the Mass Conservation Law can be applied to volumes. In this situation, the following is valid for tanks (see Figure 3.2): (3.4) where ⌬V/⌬t represents the change in volume V (m 3 ) within a time inter- val ⌬t (s). Thus, the difference between the input- and output-flow from a tank is the volume that is: 1 accumulated in the tank if Q out Ͻ Q inp (sign ϩ in Equation 3.4), 2 withdrawn from the tank if Q out Ͼ Q inp (sign Ϫ). Applied at node n that connects j pipes, the Continuity Equation can be written as: (3.5) where Q n represents the nodal discharge. An example of three pipes and a discharge point is shown in Figure 3.3. Energy Equation The Energy Equation establishes the energy balance between any two cross-sections of a pipe: (3.6)E 1 ϭ E 2 Ϯ⌬E ͚ j iϭ1 Q i Ϫ Q n ϭ 0 Q inp ϭ Q out Ϯ ⌬V ⌬t Steady Flows in Pressurised Networks 57 Q in Q out Change in stored fluid volume Figure 3.2. The Continuity Equation validity in tanks. © 2006 Taylor & Francis Group, London, UK where ⌬E is the amount of transformed energy between cross-sections 1 and 2. It is usually the energy lost from the system (the sign ϩ in Equation 3.6), but may also be added to it by pumping of water (sign Ϫ). Momentum Equation The Momentum Equation (in some literature also known as the Dynamic Equation) describes the pipe resistance to dynamic forces caused by the pressurised flow. For incompressible fluids, momentum M (N) carried across a pipe section is defined as: (3.7) where ␳ (kg/m 3 ) represents the mass density of water, Q(m 3 /s) is the pipe flow, v (m/s) is the mean velocity. Other forces in the equilibrium are (see Figure 3.4): 1 Hydrostatic force F h (N) caused by fluid pressure p (N/m 2 or Pa); F h ϭ p ϫ A. 2 Weight w (N) of the considered fluid volume (only acts in a vertical direction). 3 Force F (N) of the solid surface acting on the fluid. The Momentum Equation as written for a horizontal direction would state: (3.8) whereas in a vertical direction: (3.9) Pipe thrust The forces of the water acting on the pipe bend are the same, i.e. F x and F y but with an opposite direction i.e. a negative sign, in which case the ␳Qv 2 sin ␸ ϭϪp 2 A 2 sin ␸ ϩ w ϩ F y ␳Qv 1 Ϫ␳Qv 2 cos ␸ ϭϪp 1 A 1 ϩ p 2 A 2 cos ␸ ϩ F x M ϭ␳Qv 58 Introduction to Urban Water Distribution Q 1 Q 2 Q 1 -Q 2 +Q 3 =Q n Q 3 Q n Figure 3.3. The Continuity Equation validity in pipe junctions. © 2006 Taylor & Francis Group, London, UK total force, known as the pipe thrust will be: (3.10) The Momentum Equation is applied in calculations for the additional strengthening of pipes, in locations where the flow needs to be diverted. The results are used for the design of concrete structures required for anchoring of pipe bends and elbows. PROBLEM 3.1 A velocity of 1.2 m/s has been measured in a pipe of diameter D ϭ 600 mm. Calculate the pipe flow. Answer: The cross-section of the pipe is: which yields the flow of: PROBLEM 3.2 A circular tank with a diameter at the bottom of D ϭ 20 m and with vertical walls has been filled with a flow of 240 m 3 /h. What will be the increase of the tank depth after 15 minutes, assuming a constant flow during this period of time? Answer: The tank cross-section area is: A ϭ D 2 ␲ 4 ϭ 20 2 ϫ3.14 4 ϭ 314.16 m 2 D ϭ vA ϭ 1.2ϫ0.2827 ϭ 0.339 m 3 /sഠ340 l/s A ϭ D 2 ␲ 4 ϭ 0.6 2 ϫ3.14 4 ϭ 0.2827 m 2 F ϭ ͙F x 2 ϩ F y 2 Steady Flows in Pressurised Networks 59 p 1 A 1 p = Water pressure (N/m 2 ) A = Pipe cross-section area (m 2 ) w = Weight of the fluid (N) F = Force of the bend (N) F y w X Y F x Qv 1 p 2 A 2 Qv 2 w Figure 3.4. The Momentum Equation. © 2006 Taylor & Francis Group, London, UK The flow of 240 m 3 /h fills the tank with an additional 60 m 3 after 15 minutes, which is going to increase the tank depth by a further 60/314.16 ϭ 0.19 m Ϸ 20 cm. PROBLEM 3.3 For a pipe bend of 45° and a continuous diameter of D ϭ 300 mm, calculate the pipe thrust if the water pressure in the bend is 100 kPa at a measured flow rate of 26 l/s. The weight of the fluid can be neglected. The mass density of the water equals ␳ ϭ 1000 kg/m 3 . Answer: From Figure 3.4, for a continuous pipe diameter: Consequently, the flow velocity in the bend can be calculated as: Furthermore, for the angle ␸ ϭ 45Њ, sin ␸ ϭ cos ␸ ϭ 0.71. Assuming also that p 1 ϭ p 2 ϭ 100 kPa (or 100,000 N/m 2 ), the thrust force in the X-direction becomes: while in the Y-direction: The total force will therefore be: The calculation shows that the impact of water pressure is much more significant that the one of the flow/velocity. Self-study: Spreadsheet lesson A5.1.1 (Appendix 5) 3.1.2 Energy and hydraulic grade lines The energy balance in Equation 3.6 stands for total energies in two cross-sections of a pipe. The total energy in each cross-section comprises three components, which is generally written as: (3.11)E tot ϭ mgZ ϩ m p ␳ ϩ mv 2 2 F ϭ ͙2 2 ϩ 5 2 ഠ5.4 kN ϪF y ϭ 0.71ϫ(pA ϩ␳Qv)ഠ5 kN ϩ1000ϫ0.026ϫ0.37)ഠ2030 N ϭ 2 kN Ϫ F x ϭ 0.29ϫ(pA ϩ ␳Qv) ϭ 0.29ϫ(100,000ϫ0.07 v 1 ϭ v 2 ϭ Q A ϭ 0.026 0.07 ϭ 0.37 m/s A 1 ϭ A 2 ϭ D 2 ␲ 4 ϭ 0.3 2 ϫ3.14 4 ϭ 0.07 m 2 60 Introduction to Urban Water Distribution © 2006 Taylor & Francis Group, London, UK expressed in J or more commonly in kWh. Written per unit weight, the equation looks as follows: (3.12) where the energy obtained will be expressed in metres water column (mwc). Parameter g in both these equations stands for gravity (9.81 m/s 2 ). Potential energy The first term in Equations 3.11 and 3.12 determines the potential energy, which is entirely dependant on the elevation of the mass/volume. The second term stands for the flow energy that comes from the ability of a fluid mass m ϭ ␳ ϫ V to do work W (N) generated by the earlier- mentioned pressure forces F ϭ p ϫ A. At pipe length L, these forces create the work that can be described per unit mass as: (3.13) Kinetic energy Finally, the third term in the equations represents the kinetic energy generated by the mass/volume motion. By plugging 3.12 into 3.6, it becomes: (3.14) Bernoulli Equation In this form, the energy equation is known as the Bernoulli Equation. The equation parameters are shown in Figure 3.5. The following terminology is in common use: – Elevation head: Z 1(2) – Pressure head: p 1(2) /␳g – Piezometric head: H 1(2) ϭ Z 1(2) ϩ p 1(2) /␳g – Velocity head: v 2 1(2) /2g – Energy head: E 1(2) ϭ H 1(2) ϩ v 2 1(2) /2g The pressure- and velocity-heads are expressed in mwc, which gives a good visual impression while talking about ‘high-’ or ‘low’ pressures or energies. The elevation-, piezometric- and energy heads are compared to a reference or ‘zero’ level. Any level can be taken as a reference; it is commonly the mean sea level suggesting the units for Z, H and E in metres above mean sea level (msl). Alternatively, the street level can also be taken as a reference level. Z 1 ϩ p 1 ␳g ϩ v 1 2 2g ϭ Z 2 ϩ p 2 ␳g ϩ v 2 2 2g Ϯ⌬E W ϭ FL ϭ pAL ␳V ϭ p ␳ E tot ϭ Z ϩ p ␳g ϩ v 2 2g Steady Flows in Pressurised Networks 61 © 2006 Taylor & Francis Group, London, UK To provide a link with the SI-units, the following is valid: – 1 mwc of the pressure head corresponds to 9.81 kPa in SI-units, which for practical reasons is often rounded off to 10 kPa. – 1 mwc of the potential energy corresponds to 9.81 (Ϸ10) kJ in SI-units; for instance, this energy will be possessed by 1 m 3 of the water volume elevated 1 m above the reference level. – 1 mwc of the kinetic energy corresponds to 9.81 (Ϸ10) kJ in SI-units; for instance, this energy will be possessed by 1 m 3 of the water volume flowing at a velocity of 1 m/s. In reservoirs with a surface level in contact with the atmosphere, pressure p equals the atmospheric pressure, hence p ϭ p atm Ϸ 0. Furthermore, the velocity throughout the reservoir volume can be neglected (␷ Ϸ 0 m/s). As a result, both the energy- and piezometric-head will be positioned at the surface of the water. Hence, E tot ϭ H ϭ Z. The lines that indicate the energy- and piezometric-head levels in consecutive cross-sections of a pipe are called the energy grade line and the hydraulic grade line, respectively. The energy and hydraulic grade line are parallel for uniform flow conditions. Furthermore, the velocity head is in reality considerably smaller than the pressure head. For example, for a common pipe veloci- ty of 1 m/s, v 2 /2g ϭ 0.05 mwc, while the pressure heads are often in the order of tens of metres of water column. Hence, the real difference between these two lines is, with a few exceptions, negligible and the hydraulic grade line is predominantly considered while solving practical Energy and Hydraulic grade line 62 Introduction to Urban Water Distribution Reference level Flow direction Z 1 E 1 Energy grade lin e Hydraulic grade lin e E 2 ∆E H 2 Z 2 p 1 r r g H 1 12 p 2 g 2 g v 1 2 v 2 2 2 g Figure 3.5. The Bernoulli Equation. © 2006 Taylor & Francis Group, London, UK problems. Its position and slope indicate: – the pressures existing in the pipe, and – the flow direction. The hydraulic grade line is generally not parallel to the slope of the pipe that normally varies from section to section. In hilly terrains, the energy level may even drop below the pipe invert causing negative pressure (below atmospheric), as Figure 3.6 shows. Hydraulic gradient The slope of the hydraulic grade line is called the hydraulic gradient, S ϭ⌬E/L ϭ⌬H/L, where L (m) is the length of the pipe section. This parameter reflects the pipe conveyance (Figure 3.7). The flow rate in pipes under pressure is related to the hydraulic gra- dient and not to the slope of the pipe. More energy is needed for a pipe to convey more water, which is expressed in the higher value of the hydraulic gradient. PROBLEM 3.4 For the pipe bend in Problem 3.3 (Section 3.1.1), calculate the total energy- and piezometric head in the cross-section of the bend if it is located at Z ϭ 158 msl. Express the result in msl, J and kWh. Steady Flows in Pressurised Networks 63 Reference level Z 1 E 1 ,H 1 E 2 ,H 2 E,∆H Negative pressure Positive pressure Z 2 Figure 3.6. Hydraulic grade line. S 2 S 1 Q 1 L Q 2 E 2 E 1 Figure 3.7. The hydraulic gradient. © 2006 Taylor & Francis Group, London, UK Answer: In Problem 3.3, the pressure indicated in the pipe bend was p ϭ 100 kPa, while the velocity, calculated from the flow rate and the pipe diameter, was ␷ ϭ 0.37 m/s. The total energy can be determined from Equation 3.12: As can be seen, the impact of the kinetic energy is minimal and the difference between the total energy and the piezometric head can therefore be neglected. The same result in J and kWh is as follows: For an unspecified volume, the above result represents a type of unit energy, expressed per m 3 of water. To remember the units conversion: 1Nϭ 1kgϫ m/s 2 and 1 J ϭ 1 N ϫ m. 3.2 HYDRAULIC LOSSES The energy loss ⌬E from Equation 3.14 is generated by: – friction between the water and the pipe wall, – turbulence caused by obstructions of the flow. These causes inflict the friction- and minor losses, respectively. Both can be expressed in the same format: (3.15) Pipe resistance where R f stands for resistance of a pipe with diameter D, along its length L. The parameter R m can be characterised as a resistance at the pipe cross-section where obstruction occurs. Exponents n f and n m depend on the type of equation applied. 3.2.1 Friction losses The most popular equations used for the determination of friction losses are: 1 the Darcy–Weisbach Equation, 2 the Hazen–Williams Equation, 3 the Manning Equation. ⌬E ϭ h f ϩ h m ϭ R f Q n f ϩ R m Q n m ϭ 1650 3600 ഠ0.5 kWh E tot ϭ 168.2ϫ1000ϫ9.81 ϭ 1,650,042 Jഠ1650 kJ (or kWs) ϭ 158 ϩ 10.194 ϩ 0.007 ϭ 168.2 msl E tot ϭ Z ϩ p ␳g ϩ v 2 2g ϭ 158 ϩ 100,000 1000ϫ9.81 ϩ 0.37 2 2ϫ9.81 64 Introduction to Urban Water Distribution © 2006 Taylor & Francis Group, London, UK [...]... Bitumen/cement lined 121 129 142 137 129 142 147 69/129 — 147 142 147 125 133 145 142 133 145 149 79/ 133 — 149 145 149 130 138 147 145 — 147 150 84/ 138 147 150 147 150 132 140 150 148 — 150 152 90/140 150 152 150 152 134 141 150 148 — — — 95/141 150 1 53 150 1 53 Table 3. 3 Correction of the Hazen–Williams factors (Bhave, 1991) Chw v Ͼ 0.9 m/s per doubling less than 100 100– 130 130 –140 greater than 140 © 2006... 72 Introduction to Urban Water Distribution Answer: For a flow Q ϭ 120 l/s and a diameter of 30 0 mm, the velocity in the pipe: vϭ 4Q D2␲ ϭ 4ϫ0.12 ϭ 1.70 m/s 0 .32 3. 14 Based on the water temperature, the kinematic viscosity can be calculated from Equation 3. 22: ␷ϭ 497ϫ10Ϫ6 497ϫ10Ϫ6 ϭ ϭ 1 .31 ϫ10Ϫ6 m2/s 1.5 (T ϩ 42.5) (10 ϩ 42.5)1.5 The Reynolds number then becomes: 1.70ϫ0 .3 vD Re ϭ ␷ ϭ ϭ 3. 9ϫ105 1 .31 ϫ10Ϫ6... m/s 0.0 135 The result is substantially different than the assumed velocity and the calculation has to be continued with several more iterations The results after applying the same procedure are shown in the following table: Iter vass (m/s) D(mm) Re (-) ␭ (-) vcalc (m/s) 2 3 4 5 4.04 2.79 3. 09 3. 01 561 676 642 650 1.7 ϫ 106 1.4 ϫ 106 1.5 ϫ 106 1.5 ϫ 106 0.0141 0.0 139 0.0 139 0.0 139 2.79 3. 09 3. 01 3. 03 with... Continuity Equation (Equation 3. 5), starting from the end points of the system (Figure 3. 16) Q2 Q5 Q1+Q2 Q1–Q2+Q3+Q4+Q5 Q1 Q3+Q4 Figure 3. 16 Branched network with a single supply point © 2006 Taylor & Francis Group, London, UK Q4 Q3 88 Introduction to Urban Water Distribution If the diameters of the pipes are also known, the head-loss calculation follows the procedure in Section 3. 3.1, resulting in the hydraulic... 1ϫ1.128 vD ϭ 8.6ϫ105 Re ϭ ␷ ϭ 1 .31 ϫ10Ϫ6 © 2006 Taylor & Francis Group, London, UK 82 Introduction to Urban Water Distribution 1ϫ1.128 vD Re ϭ ␷ ϭ ϭ 8.6ϫ105 1 .31 ϫ10Ϫ6 The friction factor ␭ from Barr’s Equation equals: k ΄5.1286 ϩ 3. 7D΅ Re 0.1 5.1286 ϭ 0.25 log ΄ ϩ Ϸ 0.0 135 3. 7ϫ1128΅ (8.6ϫ10 ) / / ␭ ϭ 0.25 log2 0.89 2 5 0.89 and at Smax ϭ 0.01 the velocity from Equation 3. 28 becomes: vϭ Ί 2gDS ϭ ␭ Ί2ϫ9.81ϫ1.128ϫ0.01... 60 50 40 30 20 10 9 8 7 6 90° 3. 13 Example of minor loss diagram from valve operation 'Open' 80° 70° 60° 50° 40° 30 ° Position of disc Darcy–Weisbach Equation are: 1 length L, 2 diameter D, 3 absolute roughness k, 4 discharge Q, 5 piezometric head difference ⌬H (i.e the head-loss), 6 water temperature T © 2006 Taylor & Francis Group, London, UK 20° 10° 'Closed' 76 Introduction to Urban Water Distribution. .. Section 3. 3.2 is not recommended in this case PROBLEM 3. 8 In case the flow from the previous problem has to be doubled to Q ϭ 36 00 m3/h, calculate the diameter that would be sufficient to convey it without increasing the hydraulic gradient The other input parameters remain the same as in Problem 3. 7 (Section 3. 3.2) Answer: Assume velocity v ϭ 1 m/s Based on this velocity, the diameter D: 1 Ίv␲ ϭ 2ϫΊ1 3. 14... Qeqv =Q1 =Q2 86 Introduction to Urban Water Distribution The hydraulic calculation of the equivalent diameters further proceeds based on the principles of the single pipe calculation, as explained in Paragraph 3. 3 PROBLEM 3. 10 A pipe L ϭ 550 m, D ϭ 400 mm, and k ϭ 1 mm transports the flow of 170 l/s By an extension of the system this capacity is expected to grow to 250 l/s Two alternatives to solve this... initial velocity (usually, v ϭ 1 m/s) 2 Calculate Re from Equation 3. 21 3 Based on the Re value, choose the appropriate friction loss equation, 3. 20 or 3. 23, and calculate the ␭-factor For selected Re- and k/D values, the Moody diagram can also be used as an alternative 4 Calculate the velocity after re-writing Equation 3. 27: vϭ Ί 2gDS ␭ (3. 28) If the values of the assumed and determined velocity differ... reservoir of H ϭ 50 msl Assume for all pipes k ϭ 0.1 mm, and water temperature of 10ЊC 50 530 /250 50 410/1 1 3 2 0 /20 630 580/150 540/100 5 6 4 Nodes: ID Pipes: L(m)/D(mm) 1 Z (msl) Q (l/s) © 2006 Taylor & Francis Group, London, UK 2 3 4 5 6 — Ϫ75.6 12 10.4 22 22.1 17 10.2 25 18.5 20 14.4 90 Introduction to Urban Water Distribution Answer: The total supply from the reservoir equals the sum of all nodal . ϭ␳Qv 58 Introduction to Urban Water Distribution Q 1 Q 2 Q 1 -Q 2 +Q 3 =Q n Q 3 Q n Figure 3. 3. The Continuity Equation validity in pipe junctions. © 2006 Taylor & Francis Group, London, UK total. range of the N-values (m Ϫ1 /3 s) for typical pipe materials is given in Table 3. 4. 68 Introduction to Urban Water Distribution Table 3. 3. Correction of the Hazen–Williams factors (Bhave, 1991). C hw v. cast iron 121 125 130 132 134 Coated cast iron 129 133 138 140 141 Uncoated steel 142 145 147 150 150 Coated steel 137 142 145 148 148 Galvanised iron 129 133 — — — Uncoated asbestos cement 142 145